Electrostatics
The Electrostatic Field <--> The Electrostatic Potential
\vec{E}
V
\ \vec{E}=-\vec{\nabla} V\
\ \Delta V = -\int \vec{E}\cdot d\vec{s}\
Electrostatics
The Electric Potential
and The Electric Field
Electrostatics
\vec{E}
V
The influence
of Electric Charges
The Electric Potential
Relationship to the Electric Field
\ \Delta V = \int \vec{E}\cdot d\vec{s}\
\ \vec{E}=-\vec{\nabla} V\
Electrostatics
The Electric Potential
Relationship to the Electric Field
The Electric Field ~ the slope of the Electric Potential
\Delta V
\Delta s
\Delta V
\Delta s
Electrostatics
The Electric Potential
Relationship to the Electric Field
\vec{E}
V
\ \vec{E}=-\vec{\nabla} V\
\Delta s
\Delta V
V_1
V_2
s_1
s_2
The average Electric Field
\vec{E}_\text{avg}=-\frac{~~~~~~~~}{~~~~~~~}\ \hat{s}
\Delta V
\Delta s
magnitude = Electric Potential Difference per unit length.
direction = from High potential to Low potential
V
s
Electrostatics
The Electric Potential
Relationship to the Electric Field
V
s
s_1
s_2
V_1
V_2
\Delta s
\Delta V
\vec{E}
V
\ \vec{E}=-\vec{\nabla} V\
s_3
\vec{E}=-\ \frac{~~~~~~}{~~~~~}\ \hat{s}
d V
d s
The local Electric Field
\ \vec{E}=-\vec{\nabla} V\
In higher dimensions
Math Interlude
Differential Calculus
\vec{\nabla} f{\small (r,\theta,\phi)}= \tfrac{\partial f}{\partial r}\ \hat{r}
+
\tfrac{1}{r}\tfrac{\partial f}{\partial \theta}\ \hat{\theta}
+
\tfrac{1}{r \sin\theta}\tfrac{\partial f}{\partial \phi}\ \hat{\phi}
Spherical Coordinates
\vec{\nabla} f{\small (x,y,z)}= \tfrac{\partial f}{\partial x}\ \hat{i}
+
\tfrac{\partial f}{\partial y}\ \hat{j}
+
\tfrac{\partial f}{\partial z}\ \hat{k}
Cartesian Coordinates
\vec{\nabla} f{\small (x,y,z)}= \tfrac{\partial f}{\partial x}\ \hat{i}
+
\tfrac{\partial f}{\partial y}\ \hat{j}
+
\tfrac{\partial f}{\partial z}\ \hat{k}
Cartesian Coordinates
h{\small (x,y,z)}= ay+b
\vec{\nabla} h{\small (x,y,z)}= \tfrac{\partial h}{\partial x}\ \hat{i}
+
\tfrac{\partial h}{\partial y}\ \hat{j}
+
\tfrac{\partial h}{\partial z}\ \hat{k}
= 0\ \hat{i}
~~+~
a\ \hat{j}
~~+~
0\ \hat{k}
f{\small (x,y,z)}= axy^2+x^2 z
\vec{\nabla} f{\small (x,y,z)}=
+
x^2\ \hat{k}
+2axy\ \hat{j}
(ay^2+2xz) \hat{i}
\tfrac{\partial f}{\partial x}\ \hat{i}
+
\tfrac{\partial f}{\partial y}\ \hat{j}
+
\tfrac{\partial f}{\partial z}\ \hat{k}
=
Math Interlude
Differential Calculus
\vec{\nabla} f{\small (r,\theta,\phi)}= \tfrac{\partial f}{\partial r}\ \hat{r}
+
\tfrac{1}{r}\tfrac{\partial f}{\partial \theta}\ \hat{\theta}
+
\tfrac{1}{r \sin\theta}\tfrac{\partial f}{\partial \phi}\ \hat{\phi}
Spherical Coordinates
Math Interlude
Differential Calculus
\vec{\nabla} f{\small (r,\theta,\phi)}=
\tfrac{\partial f}{\partial r}\ \hat{r}
+
\tfrac{1}{r}\tfrac{\partial f}{\partial \theta}\ \hat{\theta}
+
\tfrac{1}{r \sin\theta}\tfrac{\partial f}{\partial \phi}\ \hat{\phi}
{\small \sin\phi}\ \hat{r}
\small +0\ \hat{\theta}
=
-
\tfrac{r\cos\phi}{r \sin\theta}\ \hat{\phi}
\vec{\nabla} V{\small (r,\theta,\phi)}=
\tfrac{\partial V}{\partial r}\ \hat{r}
+
\tfrac{1}{r}\tfrac{\partial V}{\partial \theta}\ \hat{\theta}
+
\tfrac{1}{r \sin\theta}\tfrac{\partial V}{\partial \phi}\ \hat{\phi}
=
-\tfrac{kq}{r^2}\ \hat{r}
\small +0\ \hat{\theta}
\small +0\ \hat{\phi}
\vec{E}=\tfrac{kq}{r^2}\ \hat{r}
V=\tfrac{kq}{r}
\ \vec{E}=-\vec{\nabla} V\
For a point charge
f{\small (r,\theta,\phi)}= r \sin\phi
V{\small (r,\theta,\phi)}= \tfrac{kq}{r} + V_0
Electrostatics
The Electric Potential
and The Electric Field
Electrostatics
The Electric Field
Relationship to the Electric Potential
\vec{E}
V
\ \vec{E}=-\vec{\nabla} V\
V
s
s_1
s_2
V_1
V_2
s_3
\Delta s
\Delta V
\vec{E}_\text{local}=-\frac{~~~~~~~~}{~~~~~~~}
{E}
d V
d s
\hat{s}
Electrostatics
The Electric Field
Relationship to the Electric Potential
V
s
s_1
s_2
V_1
V_2
s_3
d s
\Delta V
d s
d V
\vec{E}
V
\ \Delta V = -\int \vec{E}\cdot d\vec{s}\
\ d V = -\vec{E}\cdot d\vec{s}\
\ \Delta V = \int dV
\vec{E}_\text{local}=-\frac{~~~~~~}{~~~~~}
d V
d s
\hat{s}